How Common Is the Exponential Decay Pattern of Motor Skill Acquisition? A Brief Investigation

Abstract

Motor performance is classically described as improving nonlinearly with practice, demonstrating rapid improvements early in practice with stabilization later, which is commonly modeled by exponential decay functions. However, retrospective analyses of our previously collected data challenge this theoretical model of motor skill acquisition, suggesting that a majority of individual learners actually demonstrate patterns of motor improvement different from this classical model. A convenience sample of young adults, older adults, and people with Parkinson disease trained on the same functional upper-extremity task. When fitting three-parameter exponential decay functions to individual participant data, the authors found that only 13.3% of young adults, 40.9% of older adults, and 66.7% of adults with Parkinson disease demonstrated this “classical” skill acquisition pattern. Thus, the three-parameter exponential decay pattern may not well-represent individuals’ skill acquisition of complex motor tasks; instead, more individualized analysis methods may be warranted for advancing a theoretical understanding of motor skill acquisition.

Generally speaking, motor performance is modeled as improving nonlinearly with practice, with rapid improvements early in practice followed by eventual stabilization (e.g., Deuschl, Toro, Zeffiro, Massaquoi, & Hallett, 1996). Over the past decades, different models have sought to mathematically describe these nonlinear changes in motor performance in order to characterize skill acquisition. In particular, the power law of practice states that the logarithm of motor response time decreases linearly with the logarithm of the number of practice trials (Newell & Rosenbloom, 1981), which appeared to fit their group skill acquisition data better than a three-parameter exponential decay function. Based on this finding, it was concluded that “a single law, the power law of practice, adequately describes all of the [analyzed] practice data” (Newell & Rosenbloom, 1981). While the authors of the power law of practice were careful not to claim that all learning is log-log linear (Newell & Rosenbloom, 1981), their theory strongly influenced how skill acquisition data were analyzed in the following years, but was challenged by another seminal study showing that while the power function did in fact describe group data better than the three-parameter exponential decay function, the opposite was true when the two functions were fit to individual learners’ response time data (Heathcote, Brown, & Mewhort, 2000).

Following this, the three-parameter exponential decay function has become a common theoretical model for characterizing motor skill acquisition, both for group and individual learner data, and tasks of varying complexity (e.g., Deuschl et al., 1996; Schaefer, Dibble, & Duff, 2015; Wadden et al., 2017). Despite its pervasiveness, the three-parameter exponential decay function may not be the most appropriate method for modeling individuals’ skill acquisition in certain conditions, as our prior work demonstrated challenges with using this model to describe acquisition of a standing stepping serial reaction time task, resulting in a number of participants being excluded from our exponential decay analyses (Olivier et al., 2019). These challenges prompted us to question how prevalent this pattern of skill acquisition really is. The purpose of this quick communications report was to investigate how common the pattern of the three-parameter exponential decay function was during individual learner motor skill acquisition. To do so, a secondary analysis was performed using aggregate data previously published in papers (Paul et al., 2020; Schaefer et al., 2015; Walter, 2017) or on the Open Science Framework (https://osf.io/z4tc5/?view_only=76fee81a0cad4ab48cdc1c1f3e96b4f8), all using the same motor task paradigm among different clinical and nonclinical populations: young adults, nondemented older adults, and adults with Parkinson disease (PD).

Methods

Participants

Convenience sampling was used for this secondary analysis, and we included all cohorts who had performed the same motor task using consistent methodology as part of our prior investigations (i.e., n = 30 young adults, Walter, 2017; n = 23 nondemented older adults, Schaefer et al., 2015; plus n = 21 additional nondemented older adults—https://osf.io/z4tc5/?view_only=76fee81a0cad4ab48cdc1c1f3e96b4f8; and n = 12 adults with PD “On” levodopa, Paul et al., 2020). We included a total of 86 volunteers who were recruited through word-of-mouth or approved postings and announcements, as described in detail in the “Methods” sections of previous publications for each group (Paul et al., 2020; Schaefer et al., 2015; Walter, 2017). All participants provided informed consent prior to enrolling, and each study was approved either the University of Utah, Utah State University, or Arizona State University Institutional Review Boards. Table 1 describes characteristics of the three groups. Global cognition was measured using the Montreal Cognitive Assessment (Nasreddine et al., 2005), which is a brief cognitive screen scored from 0 to 30; higher scores indicate better performance. Stage of PD was determined using the Hoehn and Yahr scale (scored 1–5), which is a brief standardized scale describing the severity of PD symptoms; higher scores indicate worse severity (Hoehn & Yahr, 1967). Handedness was determined by participant self-report or by a modified Edinburgh Handedness Questionnaire (Oldfield, 1971), which is a multi-item questionnaire in which participants report their hand preference when performing daily functional tasks. All participants with PD were on levodopa and were taking it during training as prescribed.

Table 1.

Group Characteristics

Group	Age (years), M (SD)	Sex (F:M)	Hand dominance (R:L)	MoCA,a M (SD) [95% CI]	H&Y Stage 2,b n (%)
Young adults (n = 30)	25.1 (3.8)	20:10	29:1c	Not tested	Not applicable
Older adults (n = 44)	70.7 (10.3)	33:11	42:2d	24.9 (2.7) [24.1, 25.7]	Not applicable
PD (n = 12)	71.7 (5.1)	7:5	12:0c	26.8 (2.1) [25.5, 28.1]	12 (100%)

Note. MoCA, Montreal Cognitive Assessment; H&Y, Hoehn & Yahr; CI, confidence interval; PD = Parkinson disease.

^aMoCA is scored from 0 to 30. Scores ≥ 26 are considered normal.

^bH&Y is scored from 1 to 5. H&Y Stage 2 indicates mild-moderate PD.

^cHandedness determined by self-report.

^dHandedness determined by a modified Edinburgh Handedness Questionnaire.

Materials and Procedure

All participants completed the same motor training protocol in a single session. All participants completed 50 trials of a functional upper-extremity motor task designed to simulate aspects of self-feeding (similar to Jebsen, Taylor, Trieschmann, Trotter, & Howard, 1969). This motor task was selected because it incorporates naturalistic movements that mimic a task salient to everyday life (Schaefer, Hooyman, Duff, & Hackney, 2020) and is a feasible and efficacious method for investigating motor skill acquisition; it has been described in detail previously (Schaefer, 2015; Schaefer et al., 2015) and is shown in Figure 1 (Walter, Hengge, Lindauer, & Schaefer, 2019). A full visual description of this task can be viewed on Open Science Framework (https://osf.io/phs57/wiki/Functional_reaching_task/). Performance for each trial was calculated as trial time in seconds, in which lower values denote better performance.

Figure 1 —

Top view of the motor task. Participants used their nondominant hands to spoon two raw kidney beans at a time, as fast as possible, from a center proximal starting cup (aligned to the participant’s midline, 15 cm in front of them) to three distal target cups. All cups were secured to a board. The spoon start/end location was 5 cm lateral to the start cup. Target cups were distal to the start cup at a radius of 16 cm and at angles of 45°, 90°, and 135° relative to the start cup. The nondominant hand was used to avoid ceiling effects and allow for skill acquisition (Schaefer, 2015). Each of the 50 trials included 15 repetitions, with one repetition equal to one out-and-back movement from the starting cup to/from a target cup. Participants’ first repetition was to the ipsilateral target cup (relative to the hand used), next to the center cup, then to the contralateral cup, then back to the ipsilateral cup, completing this sequence 5 times per trial, one trial equaled 15 repetitions; a complete training session equaled 750 repetitions. Performance for each trial was calculated as trial time, defined as the amount of time required to complete 15 repetitions. Setup shown is for testing the left hand. Reprinted from “Declines in Motor Transfer Following Upper Extremity Task-Specific Training in Older Adults,” by C.S. Walter, C.R. Hengge, B.E. Lindauer, and S.Y. Schaefer, 2019, Experimental Gerontology, 116, pp. 14–19. Copyright (2019), with permission from Elsevier (Walter, Hengge, Lindauer, & Schaefer, 2019).

Statistical Analysis

The software JMP Pro 13 (SAS Institute Inc., Cary, NC) was used for statistical analyses. Individual three-parameter exponential decay functions (Equation 1) were fitted to each participant’s trial time data as a function of trial number (1–50),

$$y=\alpha +c{\text{e}}^{rx},$$ (1)

where α is the asymptote (i.e., the final trial time that the exponential decay function approaches), c is the expected change in trial time during training (i.e., from the estimated upper limit trial time to the estimated trial-time value α), r is the rate of improvement (i.e., the decay constant), and x is the trial number. R² values ≥ .25 were used to determine goodness-of-fit of this exponential decay (i.e., whether or not the participant’s motor skill acquisition pattern was well-fit by the three-parameter exponential decay function) based on published cutoff values for large effect sizes (Cohen, 1992). Descriptive statistics were then used to identify the percentage of participants in each group who demonstrated this pattern. Pearson chi-square tests were used in an exploratory analysis to determine if the observed frequency of well-fit curves in each group (i.e., young adult, older adult, and PD) differed significantly. The Open Science Framework (https://osf.io/z4tc5/?view_only=76fee81a0cad4ab48cdc1c1f3e96b4f8) can be visited for an additional exploratory analysis of whether individuals’ data were better fit by other common models.

Results

All (100%) participants in the young adult and PD groups, and 36 (81.8%) of the 44 older adult participants completed the full motor training session (50 trials); remaining participants completed 28–46 trials, but did not finish the full training session, likely due to boredom or “fatigue” (Schaefer et al., 2015). These data were, however, sufficient to be fitted with the models described above. Three-parameter exponential decay fits for each group yielded R² values of .06, .07, and .01 for the young adult, older adult, and PD groups, respectively, even though the shapes of the curves suggested exponential decreases in trial time as a function of trial number (Figure 2).

Figure 2 —

Three-parameter exponential decay fits of all data collapsed across groups: young adults (left), older adults (middle), and adults with Parkinson disease (right). Lower values on the y-axis indicate better performance. Black dots represent mean group data and grey dots represent data points for each participant. Dark lines represent the exponential decay fit.

Model fits of individual participants revealed that the three-parameter exponential decay pattern was observed in 13.3% of young adults, 40.9% of older adults, and 66.7% of adults with PD when using the R² criterion of ≥.25. Pearson chi-square tests showed that this “classical” pattern fit a significantly lower proportion of young adults (p = .05) than the sample as a whole, and they also showed a trend toward a higher proportion of adults with PD demonstrating this pattern, though it did not reach significance (p = .06). All other chi-square comparisons were nonsignificant (p > .14), indicating that there were no other significant differences in the frequency of the classical approach among the three groups of participants. The three-parameter exponential decay motor skill acquisition pattern is consistently modeled by a positive c value and a negative r value (Figure 2); however, exponential decay fits for one older adult and one adult with PD had negative c values and positive r values, thereby reflecting an atypical pattern even among those who met our preset criterion for three-parameter exponential decay.

For all other participants across groups (n = 56), the three-parameter exponential decay fit itself was not strong (R² < .25), regardless of the signs of the parameter values. Instead, a variety of different motor skill acquisition patterns were represented. Examples of some of these can be seen on the Open Science Framework (https://osf.io/z4tc5/?view_only=76fee81a0cad4ab48cdc1c1f3e96b4f8).

Because 18.2% of the older adult participants discontinued practice prior to completing the prescribed 50 practice trials, an additional analysis was performed including only the 81.8% who did complete all 50 trials. Of the 36 older adults who completed all 50 practice trials, 41.7% were well-fit by the exponential decay function (using the predetermined R² criterion of ≥.25), which is comparable to the proportion reported above for the entire older adult group (i.e., 40.9%).

Discussion

The purpose of this quick communications report was to describe how common the three-parameter exponential decay pattern actually is for motor skill acquisition. Despite widespread use of this theoretical model for analyzing motor skill acquisition data (Deuschl et al., 1996; Lang & Bastian, 1999; Olivier et al., 2019; Schaefer et al., 2015), results indicated that a pattern of rapid improvement early in training, followed by more incremental changes, leading to eventual stable performance (i.e., exponential decay), is not necessarily a representative pattern of motor skill acquisition among individuals who are young, older, or have PD. Thus, results suggest that modeling motor skill acquisition at the group or individual level with the “classical” exponential decay function may not be appropriate for capturing how individual participant performance actually changes in response to motor practice. While exponential decay nicely describes motor adaptation (e.g., visuomotor or dynamic), it may not be appropriate for more complex motor skill learning. Indeed, 33 of the 40 data sets used in the seminal study published by Heathcote were of cognitive skill acquisition, while the final seven data sets were of a simple finger-tapping serial reaction time task (Heathcote et al., 2000). Liu, Mayer-Kress, and Newell (2003) also found that their individual participants’ data sets were well-fit by an exponential decay function (and, incidentally, a power function), but their elbow angular displacement task was quite simple. In contrast, our current paper describes findings related to a naturalistic task that involved a more complex multi-joint, multi-planar movement. Thus, alternative individualized methods for quantifying change and steady-state performance may be necessary to characterize acquisition of complex motor skills beyond the “classical” approach (Park & Schweighofer, 2017). Similarly individualized approaches are used in the context of single-subject methodology (Bates, 1996) and should be considered for the purposes of modeling motor skill acquisition data, which is well-known for its variability (Stergiou & Decker, 2011).

It is of note that the three-parameter exponential decay motor skill acquisition pattern was more common in the PD group than in the other two groups, which may be somewhat surprising given the impact of this disease on motor function. We acknowledge that this sample was small, but these findings may also be due to the noted rigidity in movement strategies associated with PD, as this population is known to lack flexibility in motor strategy selection and switching (Horak, Nutt, & Nashner, 1992; Yehene, Meiran, & Soroker, 2008). Participants with PD may have chosen and persisted with a single strategy for completing the task. In contrast, participants in the other groups may have switched between strategies, resulting in other motor skill acquisition patterns.

The poor model fits shown herein could be explained by an overall lack of learning of the task. However, this possibility is unlikely given the primary outcomes presented in the initial publications from which we drew our participants. All of these primary studies included delayed retention tests to measure more durable learning of the practiced task and showed that all groups included in this secondary analysis did retain the practiced task (Paul et al., 2020; Schaefer et al., 2015; Walter, 2017; https://osf.io/z4tc5/?view_only=76fee81a0cad4ab48cdc1c1f3e96b4f8). While some of those groups (i.e., PD and older adults) performed their retention test after two additional sessions of practice that were not included in this secondary analysis (three sessions total, equaling 150 trials), the healthy young group performed only one session (50 trials total) prior to their retention test. This is noteworthy, given that the healthy young group had the lowest proportion of participants who were well-fit by exponential decay. Taken together, this suggests that an overall lack of learning is unlikely to explain the poor model fits that we found.

Limitations and Future Directions

We acknowledge that there are other measures for evaluating goodness-of-fit besides effect sizes (Spiess & Neumeyer, 2010). We also note that the R² values for each group were below our cutoff of .25, suggesting that our group data also may not have been well fit by the exponential decay model. This may be due to the specific task or effector used. However, the model fit of our group data is likely consistent with other studies using this decay function for two reasons: (a) motor skill acquisition is notoriously variable (Stergiou & Decker, 2011), which strongly influences the size of the R² values, and (b) based on visual inspection of our group scatterplots with the exponential decay lines fit to them, our participants’ data appear to have model fits comparable to those published in the literature (e.g., Schaefer et al., 2015). Unfortunately, with the exception of studies specifically aimed at testing analysis methods, past studies of motor performance and learning using exponential decay functions have: often assumed that the model would fit, rarely reported any measure of model goodness-of-fit, and tended not to fit individual subjects. As such, more work is needed to determine if other populations, tasks, and effectors also demonstrate similar variability in patterns of motor skill acquisition. If so, new theoretical approaches need to be explored for measuring motor skill acquisition and the achievement of steady-state performance beyond what has traditionally been used (Park & Schweighofer, 2017).